where \(\underline{\psi }\)  and \(\underline{A}\)  are the soft modes, and \(\overline{\psi }\)  and \(\overline{A}\)  denote the valence modes. Substituting this decomposition into the Hamiltonian  (REF ), one obtains an effective Hamiltonian for the soft-modes, including their self-interaction, and interaction of the soft modes
with the valence  [1]}. The latter are treated as classical background fields.
